Effect of flapping kinematics on aerodynamic force of a flapping two-dimensional flat plate G SENTHILKUMAR * and N R PANCHAPAKESAN Indian Institute of Technology Madras, Chennai 600 036, India e-mail: [email protected]; [email protected]; [email protected]MS received 16 March 2016; revised 6 May 2017; accepted 7 September 2017; published online 10 May 2018 Abstract. Potential applications of flapping-wing micro-aerial vehicles (MAVs) have prompted enthusiasm among the engineers and researchers to understand the flow physics associated with flapping flight. An incompressible Navier–Stokes solver that is capable of handling flapping flight kind of moving boundary problem is developed. Arbitrary Lagrangian–Eulerian (ALE) method is used to handle the moving boundaries of the problem. The solver is validated with the results of problems like inline oscillation of a circular cylinder in still fluid and a flat plate rapidly accelerating at constant angle of attack. Numerical simulations of flapping flat plate mimicking the kinematics of those like insect wings are simulated, and the unsteady fluid dynamic phenomena that enhance the aerodynamic force are studied. The solution methodology provides the velocity field and pressure field details, which are used to derive the force coefficients and the vorticity field. Time history of force coefficients and vortical structures gives insight into the unsteady mechanism associated with the unsteady aerodynamic force production. The scope of the work is to develop a computational fluid dynamic (CFD) solver with the ALE method that is capable of handling moving boundary problems, and to understand the flow physics associated with the flapping-wing aerofoil kinematics and flow parameters on aerodynamic forces. Results show that delayed stall, wing–wake interaction and rotational effect are the important unsteady mechanisms that enhance the aerodynamic forces. Major contribution to the lift force is due to the presence of leading edge vortex in delayed stall mechanism. Keywords. Insect flight; micro-aerial vehicles; flapping aerofoil; arbitrary Lagrangian–Eulerian method; unsteady forces. 1. Introduction Insects have played vital role in the design and development of micro-aerial vehicles (MAVs). Insect flight seems impossible according to the conventional aerodynamic the- ory, because to support an insect weight the wing must pro- duce two to three times more lift than that predicted by the conventional fixed wing theory [1]. Wang [2] indicated that the lift produced by flight like insect flapping is predomi- nantly higher than expected from the quasi-steady aerody- namics results. As stated in Shyy et al [3], insects have been experimenting successfully with wing design, aerodynamics, control and sensory system for millions of years. They mastered the art of flying around 350 million years ago [1]; den Berg [4] and Ellington [1] hinted that the flight like insect flapping could be a very successful design for MAVs because they have much better aerodynamic performance than con- ventional fixed-wing and rotary-wing MAVs. One of the primary design challenges in design and development of flapping-wing MAV has been the understanding of the unsteady fluid mechanics associated with the flapping wing. Early attempts to explain the force production during flap- ping flight, pioneered by Weis-Fogh [5] and Jensen [6], relied on the quasi-steady-state model, which assumes that the steady-state forces are produced by the wing at each instan- taneous position throughout a full stroke cycle. Freymuth [7] did experiments on a hovering apparatus over a limited parameter range. He also observed the thrust- producing vortical structure and calculated thrust coeffi- cient from the velocity profile of the thrust producing vortical structures. He concluded that an aerofoil in hovering can produce large thrust by full utilization of dynamic stall vortices for thrust generation. The vortical signature of this thrust is a simple vortex street with the character of a jet stream. During the early development of flapping-wing design, the time-dependent forces were correlated with the wing kinematics [8]. Recent advancement in instrumentation and high-speed video cameras provide the capability to capture the wing kinematics of flapping wing birds and insects and to measure the flow field around a flapping wing. This kinematics is being used by most of the authors for com- putational fluid dynamic (CFD) simulations to understand the flow physics. This necessitated the development of an *For correspondence 1 Sådhanå (2018) 43:72 Ó Indian Academy of Sciences https://doi.org/10.1007/s12046-018-0840-z
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Effect of flapping kinematics on aerodynamic force of a flappingtwo-dimensional flat plate
G SENTHILKUMAR* and N R PANCHAPAKESAN
Indian Institute of Technology Madras, Chennai 600 036, India
where u and v are the velocity components of the mid-
chord point in x and y directions, respectively, and T
stands for the period of oscillations. The vertical and
Figure 7. continued
72 Page 8 of 14 Sådhanå (2018) 43:72
horizontal force coefficients are obtained from the fol-
lowing equations:
Cv ¼Fy
qS; CH ¼ FX
qS
where Cv and CH are instantaneous force coefficients in x
and y directions, respectively; Fx and Fy are the forces in x
and y directions, respectively, and q ¼ 0:5qU2stands for
dynamic pressure.
5. Results and discussion
5.1 Effect of stroke deviation on force production
Insects and birds alter their wing kinematics and geom-
etry during flight to attain the efficient flight conditions.
According to the requirements, they modulate their wing
kinematics, whether to produce more lift or to fly with
better aerodynamic efficiency. The stroke deviations may
play a major role in force generation. The stroke plane
may be a horizontal stroke plane or curved stroke plane.
The presence of an attached LEV due to the delayed stall
mechanism is the most important aerodynamic mecha-
nism acting on the flapping insect wings. Though the
importance of delayed stall for horizontal stroke plane
hovering is addressed in many experimental and com-
putational studies, the significance of the same is not
directly interpreted for curved stroke plane like figure of
eight and oval shape of hovering motion. A complete
understanding of insect flight emerges only when the
wing kinematic patterns and the corresponding aerody-
namic forces acting on the wings are clear. The stroke
deviation from the mean stroke plane plays a major role
in force generation.
Stroke deviations from mean stroke plane like fig-
ure of eight, oval shape and horizontal stroke plane
kinematics are simulated and the time histories of the
force coefficients are studied. The stroke deviation used
for all the three stroke deviation cases is 0.7 times of
chord length of the aerofoil. The stroke amplitude is 1.4
times the chord length of the aerofoil. In oval pattern,
upward deviation at the start of the down stroke
requires a downward deviation at the start of the
upstroke and vice versa. In case of figure of eight pat-
tern, the two half strokes are mirror images of each
other. Horizontal stroke plane is also a kind of figure of
eight motion, but without the stroke deviation. Reynolds
number used for this simulation is 75. Kinematic
parameters aa = 45�, u = 0� and a0 = 90� are consid-
ered for the simulation.
Wang [2] addressed the effect of drag on vertical force
using inclined stroke plane hovering kinematics and con-
cluded that the drag enhances the vertical component of
the force to balance the weight of the flying object. Fig-
ure of eight motion has near-inclined stroke plane motion
during both upstroke and down stroke. During this period,
a large magnitude of drag is produced. Drag force acts
almost normal to the flat plate surface and increases the
vertical component of the force, which balances the
weight.
Even though the net vertical force produced by the
figure of eight kinematics is less than that by the hor-
izontal stroke plane kinematics, towards the end of the
figure of eight kinematics, thrust is produced. This
provides the net horizontal force to propel the flying
object; the figure of eight motion is used to produce net
horizontal force (thrust). This thrust reduces the cycle-
averaged drag coefficient. This enhances the aerody-
namic performance CV=CH
� �of the figure of eight
motion, which is shown in table 1. Figures 10(a) and
(b) shows the variation of horizontal and vertical force
coefficients with time resulting from the three kinds of
kinematics of the aerofoil. The force peaks in fig-
ures 10(a) and (b) are due to the presence of LEV
during the translation and the oscillation in the lift
coefficient curve is due to the development and the
shading of the LEV and TEV.
The trends of the horizontal stroke plane and figure of
eight motion are similar because the horizontal stroke
plane motion is also a kind of figure of eight motion,
but without the stroke deviation. The stroke-averaged
vertical, horizontal force coefficients and the ratio of
vertical to horizontal force coefficients for the three
stroke deviation cases are listed in table 1. The net
horizontal force (drag) is comparatively less in figure of
eight kinematics. The power required to overcome the
drag is less compared with the other two deviation
Figure 8. Time history of stroke parameters.
Sådhanå (2018) 43:72 Page 9 of 14 72
cases. Hence, the figure of eight motion is considered
for the parametric study in the subsequent sections.
Fruit fly and hummingbird wing kinematics are also
near figure of eight shape kinematics during hovering
motion.
5.2 Effect of amplitude of pitch oscillation on force
production
Unsteady mechanisms like delayed stall, rotational circu-
lation and wake capture are the fluid dynamic phenomena
Figure 9. (a) Horizontal stroke plane kinematics. (b) Oval shape kinematics. (c) Figure of eight kinematics, where the dot represents
the leading edge of the aerofoil.
72 Page 10 of 14 Sådhanå (2018) 43:72
that account for most of the aerodynamic force production
by flapping flight. Effect of amplitude of pitch oscillation
on the vertical and horizontal force generation is studied for
pitching angles of 45�, 30� and 15� by keeping all other
parameters the same. The change in amplitude of pitch
oscillation affects the flow attachment pattern and alters the
force coefficients. Figures 11(a) and (b) show the time
course of vertical and horizontal forces for the afore-men-
tioned pitching oscillations. These figures correspond to the
6th flapping cycle; after this the curves attain a periodic
steady-state condition. The flow structures obtained and the
time variation of force coefficients are used to study their
effect. The mean vertical force coefficient is averaged over
a complete cycle, but horizontal force coefficient is aver-
aged between the two half strokes as the flat plate changes
its direction at the end of the down stroke.
The resultant force is almost perpendicular to the flat
plate direction during the entire flapping cycle due to the
strong contribution of pressure forces. The pitch oscillation
determines the contribution of total force to vertical and
horizontal forces. The result shows that most part of the lift
is produced during down stroke, while thrust is produced at
the end of upstroke. Variation in amplitude of pitch oscil-
lation (45�, 30�, 15�) produces AoA of 135�, 120� and 105�
Table 1. Stroke-averaged vertical force coefficients for different
stroke deviations [Re = 75, A = 1.4, f = 0.1136].
Sl. no. Figure of eight Oval shape Horizontal stroke
CV 1.087 0.964 1.841
CH 1.204 1.402 2.602
CV=CH 0.902 0.678 0.707
Figure 10. (a) Horizontal force coefficient vs time history
[Re = 75, A = 1.4, f = 0.1136]. (b) Vertical force coefficient vs
time history [Re = 75, A = 1.4, f = 0.1136].
Figure 11. (a) Horizontal force coefficient vs time history
[Re = 75, A = 1.4, f = 0.1136]. (b) Vertical force coefficient vs
time history [Re 75, A = 1.4, f = 0.1136].
Sådhanå (2018) 43:72 Page 11 of 14 72
during down stroke and 45�, 60� and 75� during upstroke.
The time history of vertical force coefficient shows that the
decrease in pitch amplitude reduces the net vertical force.
However, the force peaks are the same for all the pitch
oscillations. Time history of horizontal force shows that the
pitch oscillation of 15� (AoA of 75�) does not produce
thrust towards the end of the upstroke. The stroke-averaged
vertical, horizontal force coefficients and their ratio for the
three stroke deviation cases are listed in table 2.
5.3 Effect of stroke rotation (phase angle) on force
production
Another possible means for aerodynamic force
enhancement in flapping is that the circulation around the
wing is enhanced by the quick rotation of the wing at the
end of the down stroke. Large rotational forces generated
during rotation induce a net lift force that is analogous to
the Magnus effect seen in the case of a spinning baseball.
Rotation of the leading and trailing edge leads to the
suction effect on the top surface and high pressure
stagnation area on the lower surface due to the previous
stroke vortex.
Simulations are carried out to study the effect of phase
difference between pitching and plunging angle of the
stroke on force generation. Effect of advanced (u = 30�),symmetric (u = 0�) and delayed (u = - 30�) stroke rota-
tion is studied in this section. The stroke-averaged vertical
and horizontal force coefficients are provided in table 3.
The time history of horizontal and vertical force coefficient
for one complete cycle is shown in figures 12(a) and (b).
The stroke-averaged vertical force is better during
advanced rotation (u = 30�), but the aerodynamic
performance is the best in symmetric rotation (u = 0�)case. The first peak in CV decreases from 7.3 at (u = 30�) to6.9 at (u = 0�). Similarly, the first peak in CV decreases
from 6.9 at (u = 0�) to 5.5 at (u = - 30�). Hence, the
percentage decrease from advanced to symmetrical rotation
is 7%. The percentage decrease from symmetric to delayed
rotation is 25%. Stroke-averaged vertical force coefficient
in advanced stroke is higher than in the other two cases.
However, the aerodynamic performance CV=CH
� �is the
best in symmetric rotation case, which conforms to the
results of Sane and Dickinson [15]; hence, the power
required for the symmetric rotation case will be less than
those in the other two cases. In this study the effect of phase
angle is not much pronounced because the rotation is
continuous throughout the cycle; if the rotation is restricted
to the end of the stroke, the rate of rotation will be high and
the contribution will be significant to the variation in force
coefficients.
Table 2. Stroke-averaged vertical force coefficients for different
pitch oscillations [Re = 75, A = 1.4, f = 0.1136].
A 45� 30� 15�
CV 1.087 0.756 0.569
CH 1.204 1.418 2.306
CV=CH 0.902 0.533 0.246
Table 3. Stroke-averaged vertical force coefficients for
advanced, symmetric and delayed rotation history [Re = 75,
A = 1.4, f = 0.1136].
Stroke
rotation
Advanced
rotation
Symmetric
rotation
Delayed
rotation
CV 1.392 1.205 0.621
CH 1.684 1.389 1.580
CV=CH 0.827 0.867 0.392
Figure 12. (a) Horizontal force coefficient vs time history
[Re = 75, A = 1.4, f = 0.1136]. (b) Vertical force coefficient vs
time history [Re = 75, A = 1.4, f = 0.1136].
72 Page 12 of 14 Sådhanå (2018) 43:72
5.4 Effect of Reynolds number on force production
Due to the highly inclined stroke during both the down
stroke and upstroke of figure of eight motion the vertical
force is enhanced by the drag. Wang [2] stated that
hovering motion along a horizontal stroke plane the aero-
dynamic drag does not make any contribution to the ver-
tical force. However, some of the best hover flies like
dragon flies and hoverflies employ inclined stroke plane,
where the drag during down stroke and upstroke does not
cancel each other and part of the drag enhances the vertical
force component. In this section our aim is to analyse
whether the drag mechanism acting on the figure of eight
motion can augment the vertical force production in low-
Reynolds Number flapping wings. Reynolds number was
varied from 25 to 100 in steps of 25 and the effect of Re on
force production was studied, where the Re is defined using
stroke-averaged velocity and chord length of the aerofoil.
In low-Reynolds-number flapping motion, the most com-
mon means of force generation are different from that of
the conventional flight vehicles in terms of fluid dynamic
phenomenon.
The stability of the LEV depends on the Reynolds
number. As the Re increases, the LEV is more stable and
attaches closer to the aerofoil. Hence, the core region of the
LEV comes closer to the flat plate surface and increases the
average vertical force to some extent. If we see the results
globally, there is not much difference in the average ver-
tical force coefficient. Stroke-averaged vertical force
coefficient CV for different Re is shown in table 4. Fig-
ures 13(a) and (b) show the variation of vertical force and
horizontal force coefficient with time for one complete
cycle. From the results, it is understood that the drag due to
earlier flow separation from the leading edge at low Re
does not contribute much to the vertical force coefficient.
At low Re, the LEV is unstable and quickly separates from
the top surface during the stroke reversal. Fig-
ure 13(b) shows the variation in the horizontal force coef-
ficient due to the early flow separation in low Reynolds
number. However, in case of high Re, the LEV is more
stable and attached for longer time.
6. Conclusions
The insights gained from our simulations help in under-
standing the extent of the contributions of delayed stall,
stroke reversal and wing–wake interaction in enhancement
of aerodynamic forces in flapping kinematics. The effects
of changing the wing kinematic parameters are studied and
aerodynamic force enhancement is addressed using the
vortical structures around the aerofoil. However, the results
may get altered if the three-dimensional simulations are
carried out.
At the beginning of the forward stroke, the flat plate
accelerates and pitches down. Rotation of the leading and
trailing edge leads to the suction effect on the top surface
and high pressure stagnation area on the lower surface due
to the previous stroke vortex. When the flat plate is in the
middle of the forward and backward stroke, the flat plate
moves at almost constant pitching angle; a vortex bubble is
formed on the top surface and increases the lift and drag to
their maximum value. During the translation of the flat
plate the strength of the LEV is increased due to the growth
Table 4. Effect of Re on force production for Re = 25, Re = 50,
Re = 75 and Re = 100 with A = 1.4 and f = 0.1136.
Re 25 50 75 100
CV 1.084 1.146 1.205 1.258
CH 1.200 1.281 1.389 1.457
CV=CH 0.903 0.894 0.867 0.863
Figure 13. (a) Vertical force coefficient vs time history for
Re = 25, Re = 50, Re = 75 and Re = 100 with A = 1.4,
f = 0.1136. (b) Horizontal force coefficient vs time history for
Re = 25, Re = 50, Re = 75 and Re = 100 with A = 1.4,
f = 0.1136.
Sådhanå (2018) 43:72 Page 13 of 14 72
of the vortex size. The LEV is attached to the flat plate till
the beginning of the next stroke and during the stroke
reversal the flat plate interacts with the already shed vertical
structures, which increases the effective AoA and the lift.
Hence, delayed stall, wing–wake interaction and rotational
effect are the important unsteady mechanisms enhancing
the vertical force produced by the flapping flight. A major
part of the contribution is due to the presence of LEV in
delayed stall mechanism.
The stroke deviation from the mean stroke plane plays a
major role in force generation. Figure of eight motion has
near-inclined stroke plane motion during both upstroke and
down stroke. During this period, a large magnitude of drag
is produced. Drag force acts almost normal to the flat plate
and increases the vertical component of the force, which
balances the weight. In wing kinematics like that of fig-
ure of eight, there is an imbalance in the horizontal force,
due to the thrust produced towards the end of the cycle.
Even though the stroke-averaged vertical force is higher in
horizontal stroke plane motion, the figure of eight motion
has better aerodynamic efficiency, which reduces the power
required for flight.
The change in amplitude of pitch oscillation affects the
flow attachment pattern and alters the force coefficients.
As the pitch amplitude decreases below 45�, the vertical
force as well as the aerodynamic efficiency of the fig-
ure of eight motion decreases. When the pitch oscillation
is 15�, there is no thrust at the end of the stroke. Stroke-
averaged vertical force coefficient in advanced rotation is
higher than the symmetric rotation and delayed rotation.
Hence, time of rotation plays a major role in the
enhancement of vertical force. Advanced rotation pro-
duces better vertical force but symmetric rotation pro-
vides better aerodynamic efficiency.
The stability of the LEV depends on the Reynolds
number. As the Re increases, the LEV is more stable and
attached closer to the leading edge of the flat plate.
Hence, the core region of the LEV comes closer to the
flat plate and increases the average vertical force. The
drag at low Re does not contribute much to the vertical
force coefficient.
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